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A number of exciting recent results have been seen in the field of quantumerror correction. These include initial demonstrations of error correction oncurrent quantum hardware, and resource estimates which improve understanding ofthe requirements to run large-scale quantum algorithms for real-worldapplications. In this work, we bridge the gap between these two developments byperforming careful estimation of the resources required to fault-tolerantlyperform quantum phase estimation (QPE) on a minimal chemical example.Specifically, we describe a detailed compilation of the QPE circuit to latticesurgery operations for the rotated surface code, for a hydrogen molecule in aminimal basis set. We describe a number of optimisations at both thealgorithmic and error correction levels. We find that implementing even asimple chemistry circuit requires 900 qubits and 2,300 quantum error correctionrounds, emphasising the need for improved error correction techniquesspecifically targeting the early fault-tolerant regime.

Compilation of a simple chemistry application to quantum error correction primitives